WHAT IS THE GRAPH OF $\,y = f(x)\,$?

The web exercise Graphs of Functions in the Algebra II curriculum gives a thorough introduction to graphs of functions.
For those who need only a quick review, the key concepts are repeated here.
The exercises on this current web page duplicate those in Graphs of Functions.

Understanding the Relationship between an Equation and its Graph

There are things that you can DO to an equation $\,y = f(x)\,$ that will change its graph.
Or, there are things that you can DO to a graph that will change its equation.
Stretching, shrinking, moving up/down/left/right, reflecting about axes;
they're all covered thoroughly in the next few web exercises:

An understanding of these graphical transformations makes it easy to graph a wide variety of functions,
by starting with a basic model and then applying a sequence of transformations to change it to the desired function.

For example, after mastering the graphical transformations, you'll be able to do the following:

For your convenience, all the graphical transformations are summarized in the GRAPHICAL TRANSFORMATIONS table below.
Given any entry in a row, you should (eventually!) be able to fill in all the remaining entries in that row.

SUMMARY: GRAPHICAL TRANSFORMATIONS

SET-UP FOR THE TABLE:
TRANSFORMATIONS INVOLVING $\,\boldsymbol{y}$
(note that transformations involving $\,y\,$ are intuitive)
DO THIS TO THE PREVIOUS $y$-VALUE NEW EQUATION NEW GRAPH $(a,b)$ MOVES TO ... TRANSFORMATION TYPE
add $\,p$
subtract $\,p$
$y = f(x) + p$
$y = f(x) - p$
shifts $\,p\,$ units UP
shifts $\,p\,$ units DOWN
$(a,b+p)$
$(a,b-p)$
vertical translation
vertical translation
multiply by $\,-1$ $y = -f(x)$ reflect about $x$-axis $(a,-b)$ reflection about $x$-axis
multiply by $\,g$ $y = g\cdot f(x)$ vertical stretch
by a factor of $\,g$
$(a,gb)$ vertical stretch/elongation
divide by $\,g$ $\displaystyle y = \frac{f(x)}{g}$ vertical shrink
by a factor of $\,g$
$\displaystyle \bigl(a,\frac{b}{g}\bigr)$ vertical shrink/compression
take absolute value $y = |f(x)|$ part below $x$-axis flips up $(a,|b|)$ absolute value


TRANSFORMATIONS INVOLVING $\,\boldsymbol{x}$
(note that transformations involving $\,x\,$ are counter-intuitive)
REPLACE ... NEW EQUATION NEW GRAPH $(a,b)$ MOVES TO ... TRANSFORMATION TYPE
every $\,x\,$ by $\,x+p\,$
every $\,x\,$ by $\,x-p$
$y = f(x+p)$
$y = f(x-p)$
shifts $\,p\,$ units LEFT
shifts $\,p\,$ units RIGHT
$(a-p,b)$
$(a+p,b)$
horizontal translation
horizontal translation
every $\,x\,$ by $\,-x\,$ $y = f(-x)$ reflect about $y$-axis $(-a,b)$ reflection about $y$-axis
every $\,x\,$ by $\,gx\,$ $y = f(gx)$ horizontal shrink
by a factor of $\,g$
$\displaystyle \bigl(\frac{a}{g},b\bigr)$ horizontal shrink/compression
every $\,x\,$ by $\,\displaystyle \frac{x}{g}\,$ $\displaystyle y = f\bigl(\frac{x}{g}\bigr)$ horizontal stretch
by a factor of $\,g$
$(ga,b)$ horizontal stretch/elongation
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
shifting graphs up/down/left/right
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
AVAILABLE MASTERED IN PROGRESS

 
(MAX is 16; there are 16 different problem types.)